We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in $$O(n\log ^2\!n)$$ time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejcic [CGTA’15] which uses $$O(n^{1+\delta })$$ time and $$O(n^{1+\delta })$$ space (for any constant $$\delta >0$$) and the previous randomized algorithm by Kaplan et al. [SODA’17] which uses $$O(n\log ^{12+o(1)}\!n)$$ expected time and $$O(n\log ^3\!n)$$ space. More specifically, we show that if the 2D offline insertion-only (additively) weighted nearest-neighbor problem with k operations (i.e., insertions and queries) can be solved in f(k) time, then the SSSP problem in weighted unit-disk graphs can be solved in $$O(n\log n+f(n))$$ time. Using the same framework with some new ideas, we also obtain a $$(1+\varepsilon )$$-approximate algorithm for the problem, using $$O(n\log n+n\log ^2(1/\varepsilon ))$$ time and linear space. This improves the previous $$(1+\varepsilon )$$-approximate algorithm by Chan and Skrepetos [SoCG’18] which uses $$O((1/\varepsilon )^2n\log n)$$ time and $$O((1/\varepsilon )^2 n)$$ space. More specifically, we show that if the 2D offline insertion-only weighted nearest-neighbor problem with $$k_1$$ operations in which at most $$k_2$$ operations are insertions can be solved in $$f(k_1,k_2)$$ time, then the $$(1+\varepsilon )$$-approximate SSSP problem in weighted unit-disk graphs can be solved in $$O(n\log n+f(n,O(\varepsilon ^{-2})))$$ time. Because of the $$\Omega (n\log n)$$-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

中文翻译：

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加权单位盘图中最短路径的近最优算法

我们在加权单元盘图中重新审视了一个经典的图论问题，即单源最短路径 (SSSP) 问题。我们首先提出了一种精确（和确定性）算法，该算法使用线性空间在 $$O(n\log ^2\!n)$$ 时间内解决问题，其中 n 是图的顶点数。这显着改进了 Cabello 和 Jejcic [CGTA'15] 之前的确定性算法，该算法使用 $$O(n^{1+\delta })$$ 时间和 $$O(n^{1+\delta })$$空间（对于任何常数 $$\delta > 0$$）和之前 Kaplan 等人的随机算法。[SODA'17] 使用 $$O(n\log ^{12+o(1)}\!n)$$ 预期时间和 $$O(n\log ^3\!n)$$ 空间。更具体地说，我们表明如果可以在 f(k) 时间内解决具有 k 个操作（即插入和查询）的 2D 离线仅插入（加法）加权最近邻问题，那么加权单元盘图中的SSSP问题可以在$$O(n\log n+f(n))$$时间内解决。使用相同的框架和一些新的想法，我们也得到了一个 $$(1+\varepsilon )$$-近似算法，使用 $$O(n\log n+n\log ^2(1/\varepsilon ))$$ 时间和线性空间。这改进了 Chan 和 Skrepetos [SoCG'18] 之前的 $$(1+\varepsilon )$$-近似算法，该算法使用 $$O((1/\varepsilon )^2n\log n)$$ 时间和 $$ O((1/\varepsilon )^2 n)$$ 空间。更具体地说，我们表明，如果具有 $$k_1$$ 操作（其中最多 $$k_2$$ 操作是插入的操作）的二维离线仅插入加权最近邻问题可以在 $$f(k_1,k_2)$ 中解决$ 时间，那么加权单元盘图中的 $$(1+\varepsilon )$$-近似 SSSP 问题可以在 $$O(n\log n+f(n,O(\varepsilon ^{-2 })))$$ 时间。